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Hitman42 [59]
3 years ago
6

An open rectangular box has volume 32 cm3. find the lengths of the edges giving the minimum surface area

Mathematics
1 answer:
riadik2000 [5.3K]3 years ago
7 0
Open box, so there is no top 
<span>SA = 2*l*h + 2*w*h + l*w </span>

<span>V = l*w*h = 32cm^3 </span>
<span>h = 32/(w*l) </span>

<span>SA = 2*l*32/(w*l) + 2*w*32/(w*l) + l*w </span>
<span>SA = 64/w + 64/l + l*w </span>
<span>Find minimum SA: take partial derivatives to get critical point(s) </span>
<span>SAw = -64/w^2 + l </span>
<span>SAl = -64/l^2 + w </span>
<span>Both the partials have to be 0, so... </span>
<span>0 = -64/w^2 + l and 0 = -64/l^2 + w </span>
<span>64/w^2 = l </span>
<span>0 = -64/(64/w^2)^2 + w (plug into second equation) </span>
<span>0 = -w^4/64 + w </span>
<span>0 = w(1-w^3/64) </span>
<span>1 = w^3/64 or 0 = w (impossible answer) </span>
<span>64 = w^3 </span>
<span>4 = w </span>

<span>Plug w back into 64/w^2 = l </span>
<span>64/4^2 = l </span>
<span>4 = l </span>

<span>Plug w and l back into h = 32/(w*l) </span>
<span>h = 32/(4*4) </span>
<span>h = 2 </span>

<span>The Surface Area:2*l*h + 2*w*h + l*w </span>
<span>SA = 2*4*2 + 2*4*2 + 4*4 </span>
<span>SA = 48 </span>

<span>So the answer is length = 4cm, width = 4cm, and height = 2cm, with Surface Area of 48cm^2</span>
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